A065428 - OEIS

%I #48 Jul 16 2021 13:15:18

%S 1,2,3,4,5,8,12,15,16,24,28,40,48,56,60,72,88,112,120,168,232,240,280,

%T 312,408,520,760,840,1320,1848

%N Numbers k such that no x^2 mod k is prime.

%C All numbers in this sequence except 56 are idoneal (A000926) - _Joerg Arndt_, Jul 13 2005

%C No more terms < 10^6. - _T. D. Noe_, Aug 10 2007

%C No more terms < 10^11. - _Charles R Greathouse IV_, Dec 15 2008

%C Numbers x such that all x^3 mod k are nonprimes are 1, 2, 7, 9, 63, and apparently no more.

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, p. 784

%t t={}; Do[s=Union[Mod[Range[n]^2,n]]; If[Select[s,PrimeQ]=={}, AppendTo[t,n]], {n,1000}]; t  (* _T. D. Noe_, Aug 10 2007 *)

%t nx2pQ[n_]:=Module[{m=PowerMod[Range[3n],2,n]},Count[ FindTransientRepeat[ m,2][[2]], _?PrimeQ]==0]; Select[Range[2000],nx2pQ] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jun 11 2019 *)

%o (PARI) for(n=1, 10^9, q=1; for(x=1, n-1, if(isprime(lift(Mod(x,n)^2)), q=0; break())); if(q, print1(n, ", "))); \\ edited, _Joerg Arndt_, Jan 28 2015

%o (Haskell)

%o a065428 n = a065428_list !! (n-1)

%o a065428_list = filter f [1..] where

%o    f x = all (== 0) $

%o          map (a010051' . (`mod` x) . a000290) [a000196 x .. x-1]

%o -- _Reinhard Zumkeller_, Aug 01 2012, Aug 15 2011

%o (Python)

%o from sympy import isprime

%o def ok(n): return not any(isprime((x**2)%n) for x in range(2, n))

%o print(list(filter(ok, range(1, 2000)))) # _Michael S. Branicky_, May 08 2021

%Y Cf. A179402 (x^4 mod n).

%Y Cf. A010051, A000196, A000290.

%Y Cf. A214583 (n such that for all k with gcd(n, k) = 1 and n > k^2, n - k^2 is prime).

%K nonn,nice,hard,more

%O 1,2

%A _Joerg Arndt_, Nov 16 2001